Problem: Let $f(x) = 9x^{2}-8x-3$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )?
The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $9x^{2}-8x-3 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = 9, b = -8, c = -3$ $ x = \dfrac{+ 8 \pm \sqrt{(-8)^{2} - 4 \cdot 9 \cdot -3}}{2 \cdot 9}$ $ x = \dfrac{8 \pm \sqrt{172}}{18}$ $ x = \dfrac{8 \pm 2\sqrt{43}}{18}$ $x =\dfrac{4 \pm \sqrt{43}}{9}$